In d dimensions, approximating an arbitrary function oscillating with
frequency β²k requires βΌkd degrees of freedom. A numerical
method for solving the Helmholtz equation (with wavenumber k) suffers from
the pollution effect if, as kββ, the total number of degrees of
freedom needed to maintain accuracy grows faster than this natural threshold.
While the h-version of the finite element method (FEM) (where accuracy is
increased by decreasing the meshwidth h and keeping the polynomial degree p
fixed) suffers from the pollution effect, the celebrated papers [Melenk, Sauter
2010], [Melenk, Sauter 2011], [Esterhazy, Melenk 2012], and [Melenk, Parsania,
Sauter 2013] showed that the hp-FEM (where accuracy is increased by
decreasing the meshwidth h and increasing the polynomial degree p) applied
to a variety of constant-coefficient Helmholtz problems does not suffer from
the pollution effect.
The heart of the proofs of these results is a PDE result splitting the
solution of the Helmholtz equation into "high" and "low" frequency components.
In this expository paper we prove this splitting for the constant-coefficient
Helmholtz equation in full space (i.e., in Rd) using only
integration by parts and elementary properties of the Fourier transform; this
is in contrast to the proof for this set-up in [Melenk, Sauter 2010] which uses
somewhat-involved bounds on Bessel and Hankel functions. The proof in this
paper is motivated by the recent proof in [Lafontaine, Spence, Wunsch 2022] of
this splitting for the variable-coefficient Helmholtz equation in full space;
indeed, the proof in [Lafontaine, Spence, Wunsch 2022] uses more-sophisticated
tools that reduce to the elementary ones above for constant coefficients